In the rest of this section, we borrow ideas from graph signal processing [1]} and introduce the notion of graph symmetric systems. First, we introduce a convenient way to define operations that respect the underlying communication graph structure via the graph matrix description (GMD) \(S \in \mathbb {R}^{N \times N}\) .
The matrix \(S\)  is such that the \((i,j)^{\text{th}}\)  entry is zero whenever there is no connection between components \(v_{i}\)  and \(v_{j}\) , i.e. \([S]_{ij}=0\)  if \(i \ne j \)  and \((v_{i},v_{j}) \notin \mathcal {E}\) .
Note that, since the graph is undirected, the matrix \(S\)  is symmetric.
Therefore, it has an eigedecomposition in terms of an orthonormal basis of eigenvectors \(S = V \Lambda _{S} V^{\top }\)  where \(\Lambda _{S} \in \mathbb {R}^{N \times N}\)  is a diagonal matrix with elements \(\lambda _{S,i} \in \mathbb {R}\)  such that \(S v_{i} = \lambda _{S,i} v_{i}\)  for \(v_{i} \in \mathbb {R}^{N}\)  being the \(i^{\text{th}}\)  column of \(V\) . We now introduce the notion of a graph symmetric system.
